1) Wider research context / theoretical framework This project aims at the development of a novel framework for high-order discretization of partial differential equations on general domains. The latter pose challenges related to their topology and in particular at the vicinity of, so called, extraordinary vertices where smoothness requirements and superior approximation power are paramount for efficient simulations. 2) Hypotheses/research questions /objectives We focus on the paradigm of isogeometric analysis that uses spline functions for design and analysis on non-linear geometries. We propose a framework of geometrically continuous splines called RFF-Splines (Refinable FreeForm Splines) that shall enable numerical schemes for topologically unrestricted design and analysis. 3) Approach/methods The project goes all the way from the theoretical construction to its algorithmic derivation and the efficient implementation in C++, as well as experimental evaluation in demanding applications involving high order partial differential equations. 4) Level of originality / innovation The novelty of the construction stems from the efficient construction of the basis functions (notably for evaluation and numerical integration), adaptivity by local refinement (via a truncation mechanism) as well as the good approximation power, supported by theoretical results. The idea of RFF-Splines is inspired from the work of Hartmut Prautzsch and is based on composing polynomial mappings with spline parameterizations. 5) Primary researchers involved The project involves Bert Juettler (JKU Linz), Angelos Mantzaflaris (Researcher at INRIA), Bernard Mourrain and Regis Duvigneau (Research directors at INRIA), and two PhD students (one at JKU and one at INRIA).